Convergence of cascade algorithm for individual initial function and arbitrary refinement masks (Q2574673)

For an initial function \(\phi_0\), a cascade sequence \((\phi_n)^\infty_{n=1}\) is constructed by the iteration \[ \phi_n = C_a \phi_{n-1}:= \sum_{\alpha \in \mathbb Z} a (\alpha ) \phi_{n-1} (2 \cdot - \alpha), \;n = 1,2 \dots, \] where \((a(\alpha))_{\alpha \in \mathbb Z}\) is a given finitely supported sequence on \(\mathbb Z\). The authors establish some conditions for the convergence of the cascade sequence in \(L_p (\mathbb R)\) in terms of the spectral radius of two matrices \(A_0 = (a(2\alpha - \beta))_{\alpha, \beta \in \mathbb Z}\) and \(A_1 = (a(2\alpha + 1-\beta))_{\alpha, \beta \in \mathbb Z}\) and in terms of the sequences \((\phi_1 (x+\alpha ) - \phi_0 (x+\alpha ))_{\alpha \in \mathbb Z}\) for \(x \in [0,1]\). In particular, the condition of sum rules for the mask \((a(\alpha))_{\alpha \in \mathbb Z}\) is not required. It can be proved a rate of convergence of the form \[ \| \phi_{n+1} - \phi_n \|_{L_p (\mathbb R) } = {\mathcal O}(\rho^n) \] for some \(\rho \in (0,1).\)